In this section we show that such a map can be derived, though it must be formulated as a map governing the leading order dynamics on a locally invariant \(C^r-\) smooth slow manifold \(S_\varepsilon \)  which converges to \(S\)  as \(\varepsilon \rightarrow 0\) . Under certain nondegeneracy conditions, sequential iterates of this map on the slow manifold \(S_\varepsilon \)  are \(O(\varepsilon )-\) close to each other.
We also derive the reduced \(m\) 'th iterate map on \(S_\varepsilon \)  induced by repeated iteration of (REF ). Interestingly, this map can be related to a discretization of the ODE reduced problem associated to fast-slow ODEs in [1]} if the number of iterates \(m\)  is comparable to \(\varepsilon ^{-1}\) .
